# Boundary Controllability Of Two Vibrating Strings Connected By A Point Mass With Variable Coefficients

## TL;DR

This paper extends the study of boundary controllability to two vibrating strings connected by a point mass with variable coefficients, proving exact controllability under specific conditions using spectral analysis.

## Contribution

It introduces a controllability analysis for variable coefficient strings connected by a point mass, expanding prior work with constant coefficients, and establishes spectral gap estimates.

## Key findings

- Proves exact controllability for variable coefficient strings with a point mass.
- Establishes the asymptotic behavior of spectral gaps.
- Links initial data norms to boundary control norms.

## Abstract

S. Hansen and E. Zuazua [SIAM J. Cont. Optim., 1995] studied the problem of exact controllability of two strings connected by a point mass with constant physical coefficients. In this paper we study the same problem with variable physical coefficients. This system is generated by the following equations $$\rho(x) u_{tt}=(\sigma(x) u_{x})_{x}-q(x)u,~~~~x\in (-1,0)\cup (0,1),~t>0,$$ $$Mu_{tt}(0,t)+\sigma_{1}(0)u_{x}(0^{-},t)-\sigma_{2}(0)u_{x}(0^{+},t)=0,~~~t>0,$$ with Dirichlet boundary condition on the left end and a control acts on the right end. We prove that this system is exactly controllable in an asymmetric space for the control time $T> 2\int_{-1}^{1}(\frac{\rho(x)}{\sigma(x)})^{\frac{1}{2}}dx$. We establish the equivalence between a suitable asymmetric norm of the initial data and the $L^{2}(0,T)$-norm of $u_{x}(1,t)$. Our approach is mainly based on a detailed spectral analysis and the theory of divided differences. More precisely, we prove that the spectral gap tends to zero with a precise asymptotic estimate.